NL 50, flopping a set and turn completes possible flush draw

Opp was unknown, I sat down at the table one orbit ago. I know I probably played this hand way to passive, but I simply dont know how to play a good hand OOP against an unknown when the turn completes a possible flushdraw. Lead turn and river instead of checking?

Party Poker No-Limit Hold'em, $ BB (6 handed) FTR converter on zerodivide.cx

UTG ($17.95)
MP ($54.10)
Hero ($50)
Button ($38.60)
SB ($31.30)
BB ($51.93)

Preflop: Hero is CO with 7, 7. SB posts a blind of $0.25.
2 folds, Hero raises to $2, Button calls $2, 2 folds.

Flop: ($4.75) 7, Q, 4 (2 players)
Hero bets $3.5, Button calls $3.50.

Turn: ($11.75) 9 (2 players)
Hero checks, Button bets $5, Hero calls $5.

River: ($21.75) J (2 players)
Hero checks, Button checks.

Final Pot: $21.75

Bet turn - you don't want KhX AhX or like JJ/TT/99 with a heart checking behind.

I can't really put villian on a hand here. He could have 2 hearts, AK, a Q. I can't believe he has a straight on the river (would have to call twice and bet with 8T or KT!, and if it was hearts, he'd already have a flush.).

I think you can check the river, but not intending to fold. You showed weakness on the turn, so you are inviting a bluff.

If you bet turn and he flat calls... I think then a river check is in order, or a blocking bet if you think he will only raise with the nuts.... tough river either way.